GWLR - Osun Water Points

case study : Modelling the Spatial Variation of the Explanatory Factors of Water Point Status using Geographically Weighted Logistic Regression (GWLR).

1. OVERVIEW

This study focuses on GWLR analysis based on Nigeria’s water points attributes.

1.1 Objectives

To build an explanatory model to discover factor affecting water point status in Osun State, Nigeria.

1.2 Study Area

Osun State, Nigeria

2. R PACKAGE REQUIRED

The following are the packages required for this exercise :

2.1 Load R Packages into R Environment

Usage of the code chunk below :

p_load( ) - pacman - to load packages. This function will attempt to install the package from CRAN or pacman repository list if its found not installed.

pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, ggpubr, spdep, GWmodel, tmap, skimr, caret, berryFunctions, h20)

Installing package into 'C:/Users/joeta/AppData/Local/R/win-library/4.2'
(as 'lib' is unspecified)

Warning: package 'h20' is not available for this version of R

A version of this package for your version of R might be available elsewhere,
see the ideas at
https://cran.r-project.org/doc/manuals/r-patched/R-admin.html#Installing-packages

Warning: unable to access index for repository http://www.stats.ox.ac.uk/pub/RWin/bin/windows/contrib/4.2:
  cannot open URL 'http://www.stats.ox.ac.uk/pub/RWin/bin/windows/contrib/4.2/PACKAGES'

Warning in p_install(package, character.only = TRUE, ...):

Warning in library(package, lib.loc = lib.loc, character.only = TRUE,
logical.return = TRUE, : there is no package called 'h20'

Warning in pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, ggpubr, : Failed to install/load:
h20

3. GEOSPATIAL DATA

3.1 Acquire Data Source

• Aspatial Data

  • Osun_wp_sf.rds, contained water points within Osun state.

    • It is in sf point data frame.

• Geospatial Data

  • Osun.rds, contains LGAs boundaries of Osun State.

    • It is in sf polygon data frame

3.2 Import Data

3.2.1 Import Boundary RDS File

bdy_osun <- read_rds("data/geodata/Osun.rds")

3.2.1.1 review imported data

skim(bdy_osun)

Warning: Couldn't find skimmers for class: sfc_MULTIPOLYGON, sfc; No user-
defined `sfl` provided. Falling back to `character`.

Data summary

Name	bdy_osun
Number of rows	30
Number of columns	5
_______________________
Column type frequency:
character	5
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
ADM2_EN	0	1	3	14	0	30	0
ADM2_PCODE	0	1	8	8	0	30	0
ADM1_EN	0	1	4	4	0	1	0
ADM1_PCODE	0	1	5	5	0	1	0
geometry	0	1	1805	7898	0	30	0

3.2.2 Import Attribute RDS

wp_osun <- read_rds("data/geodata/Osun_wp_sf.rds")

3.2.2.1 review imported data

skim(wp_osun)

Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.

Data summary

Name	wp_osun
Number of rows	4760
Number of columns	75
_______________________
Column type frequency:
character	47
logical	5
numeric	23
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
source	0	1.00	5	44	0	2	0
report_date	0	1.00	22	22	0	42	0
status_id	0	1.00	2	7	0	3	0
water_source_clean	0	1.00	8	22	0	3	0
water_source_category	0	1.00	4	6	0	2	0
water_tech_clean	24	0.99	9	23	0	3	0
water_tech_category	24	0.99	9	15	0	2	0
facility_type	0	1.00	8	8	0	1	0
clean_country_name	0	1.00	7	7	0	1	0
clean_adm1	0	1.00	3	5	0	5	0
clean_adm2	0	1.00	3	14	0	35	0
clean_adm3	4760	0.00	NA	NA	0	0	0
clean_adm4	4760	0.00	NA	NA	0	0	0
installer	4760	0.00	NA	NA	0	0	0
management_clean	1573	0.67	5	37	0	7	0
status_clean	0	1.00	9	32	0	7	0
pay	0	1.00	2	39	0	7	0
fecal_coliform_presence	4760	0.00	NA	NA	0	0	0
subjective_quality	0	1.00	18	20	0	4	0
activity_id	4757	0.00	36	36	0	3	0
scheme_id	4760	0.00	NA	NA	0	0	0
wpdx_id	0	1.00	12	12	0	4760	0
notes	0	1.00	2	96	0	3502	0
orig_lnk	4757	0.00	84	84	0	1	0
photo_lnk	41	0.99	84	84	0	4719	0
country_id	0	1.00	2	2	0	1	0
data_lnk	0	1.00	79	96	0	2	0
water_point_history	0	1.00	142	834	0	4750	0
clean_country_id	0	1.00	3	3	0	1	0
country_name	0	1.00	7	7	0	1	0
water_source	0	1.00	8	30	0	4	0
water_tech	0	1.00	5	37	0	20	0
adm2	0	1.00	3	14	0	33	0
adm3	4760	0.00	NA	NA	0	0	0
management	1573	0.67	5	47	0	7	0
adm1	0	1.00	4	5	0	4	0
New Georeferenced Column	0	1.00	16	35	0	4760	0
lat_lon_deg	0	1.00	13	32	0	4760	0
public_data_source	0	1.00	84	102	0	2	0
converted	0	1.00	53	53	0	1	0
created_timestamp	0	1.00	22	22	0	2	0
updated_timestamp	0	1.00	22	22	0	2	0
Geometry	0	1.00	33	37	0	4760	0
ADM2_EN	0	1.00	3	14	0	30	0
ADM2_PCODE	0	1.00	8	8	0	30	0
ADM1_EN	0	1.00	4	4	0	1	0
ADM1_PCODE	0	1.00	5	5	0	1	0

Variable type: logical

skim_variable	n_missing	complete_rate	mean	count
rehab_year	4760	0	NaN	:
rehabilitator	4760	0	NaN	:
is_urban	0	1	0.39	FAL: 2884, TRU: 1876
latest_record	0	1	1.00	TRU: 4760
status	0	1	0.56	TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
row_id	0	1.00	68550.48	10216.94	49601.00	66874.75	68244.50	69562.25	471319.00	▇▁▁▁▁
lat_deg	0	1.00	7.68	0.22	7.06	7.51	7.71	7.88	8.06	▁▂▇▇▇
lon_deg	0	1.00	4.54	0.21	4.08	4.36	4.56	4.71	5.06	▃▆▇▇▂
install_year	1144	0.76	2008.63	6.04	1917.00	2006.00	2010.00	2013.00	2015.00	▁▁▁▁▇
fecal_coliform_value	4760	0.00	NaN	NA	NA	NA	NA	NA	NA
distance_to_primary_road	0	1.00	5021.53	5648.34	0.01	719.36	2972.78	7314.73	26909.86	▇▂▁▁▁
distance_to_secondary_road	0	1.00	3750.47	3938.63	0.15	460.90	2554.25	5791.94	19559.48	▇▃▁▁▁
distance_to_tertiary_road	0	1.00	1259.28	1680.04	0.02	121.25	521.77	1834.42	10966.27	▇▂▁▁▁
distance_to_city	0	1.00	16663.99	10960.82	53.05	7930.75	15030.41	24255.75	47934.34	▇▇▆▃▁
distance_to_town	0	1.00	16726.59	12452.65	30.00	6876.92	12204.53	27739.46	44020.64	▇▅▃▃▂
rehab_priority	2654	0.44	489.33	1658.81	0.00	7.00	91.50	376.25	29697.00	▇▁▁▁▁
water_point_population	4	1.00	513.58	1458.92	0.00	14.00	119.00	433.25	29697.00	▇▁▁▁▁
local_population_1km	4	1.00	2727.16	4189.46	0.00	176.00	1032.00	3717.00	36118.00	▇▁▁▁▁
crucialness_score	798	0.83	0.26	0.28	0.00	0.07	0.15	0.35	1.00	▇▃▁▁▁
pressure_score	798	0.83	1.46	4.16	0.00	0.12	0.41	1.24	93.69	▇▁▁▁▁
usage_capacity	0	1.00	560.74	338.46	300.00	300.00	300.00	1000.00	1000.00	▇▁▁▁▅
days_since_report	0	1.00	2692.69	41.92	1483.00	2688.00	2693.00	2700.00	4645.00	▁▇▁▁▁
staleness_score	0	1.00	42.80	0.58	23.13	42.70	42.79	42.86	62.66	▁▁▇▁▁
location_id	0	1.00	235865.49	6657.60	23741.00	230638.75	236199.50	240061.25	267454.00	▁▁▁▁▇
cluster_size	0	1.00	1.05	0.25	1.00	1.00	1.00	1.00	4.00	▇▁▁▁▁
lat_deg_original	4760	0.00	NaN	NA	NA	NA	NA	NA	NA
lon_deg_original	4760	0.00	NaN	NA	NA	NA	NA	NA	NA
count	0	1.00	1.00	0.00	1.00	1.00	1.00	1.00	1.00	▁▁▇▁▁

3.3 Exploratory Data Analysis (EDA)

3.3.1 Plot Bar Chart

3.3.1.1 visualise “status”

wp_osun %>%
  freq(input = "status")

Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
  Please report the issue at <https://github.com/pablo14/funModeling/issues>.

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0

3.3.1.2 visualise “status” by “water_tech_category”

cross_plot(data = wp_osun, input = "water_tech_category", target = "status")

3.3.1.3 visualise “status” by “usage_capacity”

cross_plot(data = wp_osun, input = "usage_capacity", target = "status")

3.3.2 Visualise Distribution of “status” Variable

tmap_mode("view")

tmap mode set to interactive viewing

tm_shape(bdy_osun) +
  tm_polygons(alpha = 0.4) +
  
tm_shape(wp_osun) +
  tm_dots(col = "status",
          alpha = 0.6) +
  tm_view(set.zoom.limits = c(8.5,12))

tmap_mode("plot")

tmap mode set to plotting

3.4 Data Wrangling

3.4.1 Edit Key Variables

wp_osun.sf <- wp_osun %>%
  filter_at(vars(status,
                 distance_to_primary_road,
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city,
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean),
            all_vars(!is.na(.)))%>%
  mutate(usage_capacity = as.factor(usage_capacity))

Remarks :

Convert “usage_capacity” from numeric to categorical variable via as.factor( ) function.

3.4.2 Get Column Index

match(c("distance_to_primary_road",
        "distance_to_secondary_road",
        "distance_to_tertiary_road",
        "distance_to_city",
        "distance_to_town",
        "water_point_population",
        "local_population_1km",
        "usage_capacity",
        "is_urban",
        "water_source_clean",
        "status",
        "geometry"),
      names(wp_osun.sf))

 [1] 35 36 37 38 39 42 43 46 47  7 57 NA

3.4.3 Create Correlation Analysis Data Table

wp_osun.sf_clean <- wp_osun.sf %>%
  select(c(7, 35:39, 42:43, 46:47, 57)) %>%
  st_set_geometry(NULL)

4. CORRELATION ANALYSIS

4.1 Visualise Correlation Matrix

cluster_vars.cor = cor(wp_osun.sf_clean[,2:7])

corrplot.mixed(cluster_vars.cor,
              lower = "ellipse",
              upper = "number",
              tl.pos = "lt",
              diag = "l",
              tl.col = "black")

Remarks :

The correlation matrix above indicated there is no highly correlated variable pairs.

5. LOGISTIC REGRESSION MODEL

5.1 Build Logistic Regression Model

model <- glm(status ~
               distance_to_primary_road +
               distance_to_secondary_road +
               distance_to_tertiary_road +
               distance_to_city +
               distance_to_town +
               is_urban +
               usage_capacity +
               water_source_clean +
               water_point_population +
               local_population_1km,
             data = wp_osun.sf,
             family = binomial(link = 'logit'))

5.1.1 Create Model Overview :: model

blr_regress(model)

                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------

5.1.2 Generate Confusion Matrix :: model

blr_confusion_matrix(model, cutoff = 0.5)

Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

Remarks :

For a logistic regression model, the specificity rate is only 0.6154.

6. GEOGRAPHICALLY WEIGHTED LOGISTIC REGRESSION (GWLR) MODEL

6.1 Convert Simple Feature to SpatialPointsDataFrame

wp_osun.sp <- wp_osun.sf %>%
  select(c(status,
           distance_to_primary_road,
           distance_to_secondary_road,
           distance_to_tertiary_road,
           distance_to_city,
           distance_to_town,
           is_urban,
           usage_capacity,
           water_source_clean,
           water_point_population,
           local_population_1km)) %>%
  as_Spatial()

wp_osun.sp

class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, is_urban, usage_capacity, water_source_clean, water_point_population, local_population_1km 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,        0,           1000,           Borehole,                      0,                    0 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,        1,            300,   Protected Spring,                  29697,                36118 

6.2 Fixed Bandwidth GWR Model

6.2.1 Compute Fixed Bandwidth

bw.fixed <- bw.ggwr(status ~ distance_to_primary_road +
                      distance_to_secondary_road +
                      distance_to_tertiary_road +
                      distance_to_city +
                      distance_to_town +
                      is_urban +
                      usage_capacity +
                      water_source_clean +
                      water_point_population +
                      local_population_1km,
                    data = wp_osun.sp,
                    family = "binomial",
                    approach = "AIC",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)

bw.fixed

Remarks :

Recommended bandwidth is 2,599.672 metres.

6.2.2 Perform Fixed Bandwidth GWR Model

gwlr.fixed <- ggwr.basic(status ~
                           distance_to_primary_road + 
                           distance_to_secondary_road +
                           distance_to_tertiary_road +
                           distance_to_city +
                           distance_to_town +
                           is_urban +
                           usage_capacity +
                           water_source_clean +
                           water_point_population +
                           local_population_1km,
                    data = wp_osun.sp,
                    bw = 2599.672,
                    family = "binomial",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)

 Iteration    Log-Likelihood
=========================
       0        -1958 
       1        -1676 
       2        -1526 
       3        -1443 
       4        -1405 
       5        -1405 

Remarks :

Calibrate the model with the recommended bandwidth, bw = 2,599.672.

gwlr.fixed

   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:30:16 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    is_urban + usage_capacity + water_source_clean + water_point_population + 
    local_population_1km, data = wp_osun.sp, bw = 2599.672, family = "binomial", 
    kernel = "gaussian", adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
water_point_population                   ***
local_population_1km                     ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2599.672 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.7229e+02 -4.9955e+00  1.7600e+00
   distance_to_primary_road                 -1.9389e-02 -4.8031e-04  2.9618e-05
   distance_to_secondary_road               -1.5921e-02 -3.7551e-04  1.2317e-04
   distance_to_tertiary_road                -1.5618e-02 -4.2368e-04  7.6179e-05
   distance_to_city                         -1.8416e-02 -5.6217e-04 -1.2726e-04
   distance_to_town                         -2.2411e-02 -5.7283e-04 -1.5155e-04
   is_urbanTRUE                             -1.9790e+02 -4.2908e+00 -1.6864e+00
   usage_capacity1000                       -2.0772e+01 -9.7231e-01 -4.1592e-01
   water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01  5.3340e-01
   water_source_cleanProtected.Spring       -5.2235e+02 -5.5977e+00  2.5441e+00
   water_point_population                   -5.2208e-02 -2.2767e-03 -9.8875e-04
   local_population_1km                     -1.2698e-01  4.9952e-04  1.0638e-03
                                                3rd Qu.      Max.
   Intercept                                 1.2763e+01 1073.2156
   distance_to_primary_road                  4.8443e-04    0.0142
   distance_to_secondary_road                6.0692e-04    0.0258
   distance_to_tertiary_road                 6.6815e-04    0.0128
   distance_to_city                          2.3718e-04    0.0150
   distance_to_town                          1.9271e-04    0.0224
   is_urbanTRUE                              1.2841e+00  744.3099
   usage_capacity1000                        3.0322e-01    5.9281
   water_source_cleanProtected.Shallow.Well  1.7849e+00   67.6343
   water_source_cleanProtected.Spring        6.7663e+00  317.4133
   water_point_population                    5.0102e-04    0.1309
   local_population_1km                      1.8157e-03    0.0392
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2795.084 
   AIC : 4414.606 
   AICc : 4747.423 
   Pseudo R-square value:  0.5722559 

   ***********************************************************************
   Program stops at: 2022-12-18 00:31:11 

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

6.2.3 Set Threshold Value

• Set this exercise’s threshold value, otherwise known as “yhat” to 0.5.

• The value will be assigned to 1 when greater than 0.5, else 0, and saved under the “most” variable.

gwr.fixed <- gwr.fixed %>% 
  mutate(most = ifelse(
    gwr.fixed$yhat >= 0.5, T, F
  ))

6.2.3.1 visualise “most”

freq(gwr.fixed$most)

Warning in freq(gwr.fixed$most): All input values are NA.

NULL

6.2.4 Generate Confusion Matrix

gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data = gwr.fixed$most, reference = gwr.fixed$y)
CM

Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.8628          
            Specificity : 0.9005          
         Pos Pred Value : 0.8740          
         Neg Pred Value : 0.8913          
             Prevalence : 0.4445          
         Detection Rate : 0.3835          
   Detection Prevalence : 0.4388          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : FALSE           
                                          

6.2.5 Visualise GWLR

6.2.5.1 extract administrative variables

wp_osun.sf_selected <- wp_osun.sf %>% 
  select(c(ADM2_EN, ADM2_PCODE, ADM1_EN, ADM1_PCODE, status))

6.2.5.2 combine wp_osun.sf_selected and gwr.fixed

gwr_sf.fixed <- cbind(wp_osun.sf_selected, gwr.fixed)

6.2.5.3 visualise coefficient estimates

tmap_mode("view")

tmap mode set to interactive viewing

prob_T <- tm_shape(bdy_osun) +
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
  tm_dots(col = "yhat",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(8.5,14))
prob_T

tmap_mode("plot")

tmap mode set to plotting

7 MODEL CALIBRATION

7.1 Calibrate Logistic Regression Model

Remove the insignificant variables “distance_to_primary_road” and “distance_to_secondary_road” that identified under section 5.2.

model_calibr <- glm(status ~  distance_to_tertiary_road+
                 distance_to_city+
                 distance_to_town+
                 water_point_population+
                 local_population_1km+
                 usage_capacity+
                 is_urban+
                 water_source_clean,
             data = wp_osun.sf,
             family = binomial(link = "logit"))

7.2 Create Model Overview :: model_calibr

blr_regress(model_calibr)

                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4746           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3540        0.1055      3.3541       8e-04 
       distance_to_tertiary_road            1      1e-04         0.0000      4.9096      0.0000 
            distance_to_city                1      0.0000        0.0000     -5.2022      0.0000 
            distance_to_town                1      0.0000        0.0000     -5.4660      0.0000 
         water_point_population             1      -5e-04        0.0000    -11.3902      0.0000 
          local_population_1km              1      3e-04         0.0000     19.4069      0.0000 
           usage_capacity1000               1     -0.6206        0.0697     -8.9081      0.0000 
              is_urbanTRUE                  1     -0.2667        0.0747     -3.5690       4e-04 
water_source_cleanProtected Shallow Well    1      0.4947        0.0850      5.8228      0.0000 
   water_source_cleanProtected Spring       1      1.2790        0.4384      2.9174      0.0035 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7349          Somers' D        0.4697   
% Discordant          0.2651          Gamma            0.4697   
% Tied                0.0000          Tau-a            0.2320   
Pairs                5585188          c                0.7349   
---------------------------------------------------------------

7.3 Generate Confusion Matrix :: model_calibr

blr_confusion_matrix(model_calibr, cutoff = 0.5)

Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1300  743
         1   814 1899

                Accuracy : 0.6726 
     No Information Rate : 0.4445 

                   Kappa : 0.3348 

McNemars's Test P-Value  : 0.0761 

             Sensitivity : 0.7188 
             Specificity : 0.6149 
          Pos Pred Value : 0.7000 
          Neg Pred Value : 0.6363 
              Prevalence : 0.5555 
          Detection Rate : 0.3993 
    Detection Prevalence : 0.5704 
       Balanced Accuracy : 0.6669 
               Precision : 0.7000 
                  Recall : 0.7188 

        'Positive' Class : 1

Remarks :

The specificity rate, 0.6149 is slightly lower than the original logistic regression model, which is 0.6154.

wp_osun.sp2 <- wp_osun.sf %>%
  select(c(status,
           distance_to_tertiary_road,
           distance_to_city,
           distance_to_town,
           water_point_population,
           local_population_1km,
           usage_capacity,
           is_urban,
           water_source_clean)) %>%
  as_Spatial()

wp_osun.sp2

class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 9
names       : status, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean 
min values  :      0,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,           1000,        0,           Borehole 
max values  :      1,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,            300,        1,   Protected Spring 

7.4 Fixed Bandwidth GWR Model :: model_calibr

bw.fixed_calibr <- bw.ggwr(status ~ distance_to_tertiary_road+
                 distance_to_city+
                 distance_to_town+
                 water_point_population+
                 local_population_1km+
                 usage_capacity+
                 is_urban+
                 water_source_clean,
             data = wp_osun.sp2,
             family = 'binomial',
             approach = 'AIC',
             kernel = 'gaussian',
             adaptive = FALSE,
             longlat = FALSE)

bw.fixed_calibr

Remark :

The recommended bandwidth is 2,377.371 metres.

7.4.1 Compute Fixed Bandwidth

gwlr.fixed_calibr <- ggwr.basic(status ~ distance_to_tertiary_road +
                                  distance_to_city +
                                  distance_to_town +
                                  water_point_population +
                                  local_population_1km +
                                  usage_capacity +
                                  is_urban +
                                  water_source_clean,
                                data = wp_osun.sp2,
                                bw = 2377.371,
                                family = 'binomial',
                                kernel = 'gaussian',
                                adaptive = FALSE,
                                longlat = FALSE)

 Iteration    Log-Likelihood
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 

gwlr.fixed_calibr

   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:31:13 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + water_point_population + local_population_1km + 
    usage_capacity + is_urban + water_source_clean, data = wp_osun.sp2, 
    bw = 2377.371, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
usage_capacity1000                       ***
is_urbanTRUE                             ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2377.371 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -3.7021e+02 -4.3797e+00  3.5590e+00
   distance_to_tertiary_road                -3.1622e-02 -4.5462e-04  9.1291e-05
   distance_to_city                         -5.4555e-02 -6.5623e-04 -1.3507e-04
   distance_to_town                         -8.6549e-03 -5.2754e-04 -1.6785e-04
   water_point_population                   -2.9696e-02 -2.2705e-03 -1.2277e-03
   local_population_1km                     -7.7730e-02  4.4281e-04  1.0548e-03
   usage_capacity1000                       -5.5889e+01 -1.0347e+00 -4.1960e-01
   is_urbanTRUE                             -7.3554e+02 -3.4675e+00 -1.6596e+00
   water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01  6.2378e-01
   water_source_cleanProtected.Spring       -1.3630e+03 -5.3436e+00  2.7714e+00
                                                3rd Qu.      Max.
   Intercept                                 1.3755e+01 2171.6375
   distance_to_tertiary_road                 6.3011e-04    0.0237
   distance_to_city                          1.5921e-04    0.0162
   distance_to_town                          2.4490e-04    0.0179
   water_point_population                    4.5879e-04    0.0765
   local_population_1km                      1.8479e-03    0.0333
   usage_capacity1000                        3.9113e-01    9.2449
   is_urbanTRUE                              1.0554e+00  995.1841
   water_source_cleanProtected.Shallow.Well  1.9564e+00   66.8914
   water_source_cleanProtected.Spring        7.0805e+00  208.3749
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2815.659 
   AIC : 4418.776 
   AICc : 4744.213 
   Pseudo R-square value:  0.5691072 

   ***********************************************************************
   Program stops at: 2022-12-18 00:32:04 

7.4.2 Convert Data Frame

gwr.fixed_calibr <- as.data.frame(gwlr.fixed_calibr$SDF)

7.4.3 Set Threshold Value

gwr.fixed_calibr <- gwr.fixed_calibr %>%
  mutate(most = ifelse(
    gwr.fixed_calibr$yhat >= 0.5, T, F))

7.4.4 Generate Confusion Matrix

gwr.fixed_calibr$y <- as.factor(gwr.fixed_calibr$y)
gwr.fixed_calibr$most <- as.factor(gwr.fixed_calibr$most)
CM <- confusionMatrix(data = gwr.fixed_calibr$most, 
                      reference = gwr.fixed_calibr$y)
CM

Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1833  268
     TRUE    281 2374
                                          
               Accuracy : 0.8846          
                 95% CI : (0.8751, 0.8935)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7661          
                                          
 Mcnemar's Test P-Value : 0.6085          
                                          
            Sensitivity : 0.8671          
            Specificity : 0.8986          
         Pos Pred Value : 0.8724          
         Neg Pred Value : 0.8942          
             Prevalence : 0.4445          
         Detection Rate : 0.3854          
   Detection Prevalence : 0.4418          
      Balanced Accuracy : 0.8828          
                                          
       'Positive' Class : FALSE           
                                          

7.5 Visualise GWLR

wp_osun.sf_selected_calibr <- wp_osun.sf %>%
  select(c(ADM2_EN, ADM2_PCODE, ADM1_EN, ADM1_PCODE, status))

gwr_sf.fixed_calibr <- cbind(wp_osun.sf_selected_calibr, gwr.fixed_calibr)

7.6 Visualise Functional & Non-Functional Water Point

tmap_mode("view")

tmap mode set to interactive viewing

prob_TCalibr <- tm_shape(bdy_osun) + 
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed_calibr) + 
  tm_dots(col = "yhat",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(9,14))

prob_TCalibr

tmap_mode("plot")

tmap mode set to plotting

8 REFERENCE

Chua A. (2022). In-class Ex5: Modelling the Spatial Variation of the Explanatory Factors of Water Point Status using Geographically Weighted Logistic Regression. https://isss624-amelia.netlify.app/exercises/in-class_ex5/in-class_ex5